 # It is a common observation that rain clouds `
Question:

It is a common observation that rain clouds can be at about a kilometre altitude above the ground.

(a) If a rain drop falls from such a height freely under gravity, what will be its speed? Also, calculate in km/h

(b) A typical rain drop is about 4 mm diameter. Momentum is mass x speed in magnitude. Estimate its momentum when it hits ground.

(c) Estimate the time required to flatten the drop.

(d) Rate of change of momentum is force. Estimate how much force such a drop would exert on you.

(e) Estimate the order of magnitude force on umbrella. Typical lateral separation between two rain drops is 5 cm.

Solution:

(a) Velocity attained by the rain drop which is falling freely through the height h is:

v2 = u2 – 2g(-h)

As u = 0

v = √2gh = 100√2 m/s = 510 km/h

(b) Diameter of the drop, d = 2r = 4 mm

Radius of the drop, r = 2 mm = 2 × 10-3 m

Mass of the drop, m = Vρ = 3.4 × 10-5 kg

Momentum of the rain drop, p = mv = 5 × 10-3 kg.m/s

(c) Time required to flatten the drop is the time taken by the drop to reach the ground

t = d/v = 30 ms

(d) Force exerted by a rain drop is, F = change in momentum/time = p-0/t = 168 N

(e) Radius of umbrella, R = 1/2 m

Area of umbrella = 0.8 m2

No.of drops striking the umbrella, n = 320

Therefore the net force exerted = 54000N